An M × M covariance matrix R exhibits many special properties. For example, it is complex Hermitian, equal to its conjugate transpose, RH = R; it is positive semidefinite, xHRx ≥ 0. Because of the latter, its eigenvalues are greater than or equal to zero as well. In many cases, it is also Toeplitz Ri,j = Ri + m,j + m, that is, diagonal elements are equal. In this chapter, I will review some of the more important properties of covariance matrices and their eigenstructure, and discuss some simple applications.
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This reference is intended to give the geophysical signal analyst sufficient material to understand the usefulness of data covariance matrix analysis in the processing of geophysical signals. A background of basic linear algebra, statistics, and fundamental random signal analysis is assumed. This reference is unique in that the data vector covariance matrix is used throughout. Rather than dealing with only one seismic data processing problem and presenting several methods, we will concentrate on only one fundamental methodology—analysis of the sample covariance matrix—and we present many seismic data problems to which the methodology applies.